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MANAGEMENT BRIEF 203 ASMR 3 ASMR 3 differs from formulations 1 and 2 in that more flexibility is allowed in the estimation of age- and time:specific capture probabilities (fi a)' As in ASMR 2, U T is estimated directly; however, we use the conditional maximum likelihood estimate for the P a,l matrix, that is, ^ rna,l + ra.t Pa.t = ^. ^ . . Ua,t + Ma,t This formulation eliminates the need to specify annual vulnerability schedules at the expense of a much larger parameter set. ASMR 3 maximizes equation (6) by varying e = (M'adull and Ua.T). One strength of the overall ASMR approach is that it allows the estimation of recruit abundance for years preceding the onset of data collection by using age- specific survival rates and initial-year abundance as follows: ^ Ua.l U2,2-a = ~. nSj t=2 This is an important aspect of the ASMR and a distinction from the "recruitment" parameter esti- mated by Jolly-Seber type methods (discussed below). Assignment of Apparent Age at First Capture We reparameterized the polynomial length-at-age relationship contained in the humpback chub Endan- gered Species Act recovery goals and based on fish collected in Grand Canyon (USFWS 2002) to a von Bertalanffy growth function necessary for the Lorenzen mortality curve. We then used the inverse von Bertalanffy growth function to assign apparent age as a function of length, that is, a=-~log,,(I-f)-ao, (11) where k=0.12, ao =0.87, and I", =455. We conducted sensitivity analyses on the effect of growth parameter misspecification on abundance and recruitment trends. Model Performance We tested the estimation methods by generating simulated capture-recapture data from known numbers of fish Ua,l and U2,t from ages 2-30 and years 1989- 2002 subject to individual binomial capture, survival, and recapture events over age and time with known va.1 and annual capture probabilities (fi/)' These tests indicate that the method gives unbiased estimates of the PI' Ua,l up to at least age 10, the U2,1 for all t, Va,1 for the time periods mentioned above, and age-specific survival rates Sa up to at least a = 20. The simulated (9) recruitment and initial stock estimates U2,t and Ua.l are precise (<10% error) for the period from the early to the mid 1990s and become imprecise for the late 1990s to 2002. Most S estimates are precise for all a. a We cannot assign an accurate age to each humpback chub at first capture. The recapture size data indicate that growth rates are extremely variable; for example, the average size of an age-l0 humpback chub is around 300 mm TL, but fish of this size can be anywhere between about 6 and 15 years of age. When we assign a fish an "age" at first capture, that age may well be predictive of the subsequent size-dependent survival rate (Lorenzen model) but is probably not the correct age for assigning the fish to a cohort. This means that the y,7ar-class strengths or apparent cohort sizes Ua,l and U2,1 estimated by the procedure outlined above are not the numbers of fish recruiting or initially present by cohort but rather some smoothed or running average of the actual cohort strengths. The estimation method should still be able to detect longer-term trends in abundance and recruitment, but it is less able to detect subtle changes in year-class strength. The use of VP A back propagation for calculating Ua,1 does not cause the ASMR method to overestimate adult abundance. To evaluate this, we replaced equation (3) with the forward prediction equation (1) for U and A 0.1 included th,7 initial unmarked abundances Ua,l (a = 2, . . .,30) and U2,I (t > 1) in the set ofunIrnown parameters to be estimated by maximizing equation (6). This resulted in essentially the same abundance and adult mortality estimates as the back propagation method. While the proposed estimation method is unbiased when supplied with accurate ages at first capture, tests with simulated data lhat include initial aging error indicate lhat it produces estimates of A. and M'adUlt that are biased downward by about 7% and 11 %, re- spectively. However, it still tracks the simulated recruitment patterns over time and does not create a spurious simulated trend in back-calculated recruit- ments before lhe first year of sampling (Figure I). As a measure of uncertainty, we sampled lhe posterior distribution of the parameter estimates using Markov chain-Monte Carlo (MCMC) techniques for each ASMR formulation. Following Gelman et al. (2000), we constructed 95%-credible intervals from MCMC parameter chains of length 1,000 from 200,000 MCMC trials, retaining every l00th trial and disre- garding the first half of lhe chain. We assessed convergence using Gelman and Rubin's potential scale reduction factor (R Development Core Team 2005). Discussion The abundance and mortality trends from lhe ASMR and Jolly-Seber models are similar (Coggins et al. (10)